Locally Asplund preduals of spaces of holomorphic functions (Q1872920)

This paper on Banach spaces and linear topology studies versions of the Radon--Nikodým property, and applies them to spaces of holomorphic functions. The theorems are as follows. Theorem~1. A Banach space \(E\) with a boundedly complete Schauder decomposition \((E_n)_{n=1}^\infty\) has the Radon--Nikodým property if and only if each \(E_n\) has. Theorem~2. If a Banach space \(E\) has an unconditional Schauder decomposition \((E_n)_{n=1}^\infty\) with each \(E_n\) Asplund, then the following are equivalent. (1) \(E\) is Asplund. (2) \(E\) does not contain \(\ell_1\). (3) \((E_n)\) is a shrinking decomposition. (4) \(E'_b\) does not contain \(c_0\). Theorem~9. Let \(E\) be a Banach space, \({\mathcal P}(^nE)\) the space of holomorphic polynomials \(E\to{\mathbb C}\) of degree \(n\). Then \({\mathcal P}(^nE)\) has the Radon--Nikodým property for all \(n\) if and only if for any choice of \(f:U\to F\), where \(f\) is an integral function of bounded type, \(U\subset E\) balanced open, \(F\) a Banach space, \(f\) is also nuclear. Here \(f\) is nuclear if it is holomorphic of bounded type, and it can be written as a certain type of series with numerical coefficients in \(\ell_1\); \(f\) is integral if it is holomorphic of bounded type, and its `component functions' can be written as a certain type of integral. The proofs are short, technical, and draw heavily upon the literature. Despite some typos the paper is very readable and informative.